Question 12 Determine the slope of the following function at the point $(-2,1)$. \[ x^{2} y+y^{4}=3+2 y \] a. $2 / 3$ b. $3 / 2$ c. $-3 / 2$ d. 1

Step 1 ：First, we need to find the derivative of the given function. To do this, we will use the implicit differentiation method.

Step 2 ：The given function is \(x^{2} y+y^{4}=3+2 y\). Differentiating both sides with respect to \(x\), we get \(2xy + x^{2}y' + 4y^{3}y' = 2y'\).

Step 3 ：Rearranging the terms, we get \(x^{2}y' + 4y^{3}y' - 2y' = -2xy\).

Step 4 ：Factoring out \(y'\), we get \(y'(x^{2} + 4y^{3} - 2) = -2xy\).

Step 5 ：Solving for \(y'\), we get \(y' = \frac{-2xy}{x^{2} + 4y^{3} - 2}\).

Step 6 ：Now, we substitute the point \((-2,1)\) into the derivative to find the slope of the function at this point.

Step 7 ：Substituting, we get \(y' = \frac{-2(-2)(1)}{(-2)^{2} + 4(1)^{3} - 2} = \frac{4}{4 + 4 - 2} = \frac{4}{6} = \frac{2}{3}\).

Step 8 ：So, the slope of the function at the point \((-2,1)\) is \(\frac{2}{3}\).

Step 9 ：Therefore, the answer is (a) \(\frac{2}{3}\).

Step 10 ：\boxed{\frac{2}{3}}